In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles. While sometimes called an exchange force, or, in the case of fermions, Pauli repulsion, its consequences cannot always be predicted based on classical ideas of force. Both bosons and fermions can experience the exchange interaction.
The wave function of indistinguishable particles is subject to exchange symmetry: the wave function either changes sign (for fermions) or remains unchanged (for bosons) when two particles are exchanged. The exchange symmetry alters the expectation value of the distance between two indistinguishable particles when their wave functions overlap. For fermions the expectation value of the distance increases, and for bosons it decreases (compared to distinguishable particles).David J. Griffiths: Introduction to Quantum Mechanics, Second Edition, pp. 207–210
The exchange interaction arises from the combination of exchange symmetry and the Coulomb interaction. For an electron in an electron gas, the exchange symmetry creates an "exchange hole" in its vicinity, which other electrons with the same spin tend to avoid due to the Pauli exclusion principle. This decreases the energy associated with the Coulomb interactions between the electrons with same spin. Since two electrons with different spins are distinguishable from each other and not subject to the exchange symmetry, the effect tends to align the spins. The exchange interaction is the main physical effect responsible for ferromagnetism, and has no classical analogue.
For bosons, the exchange symmetry makes them bunch together, and the exchange interaction takes the form of an effective attraction that causes identical particles to be found closer together, as in Bose–Einstein condensation.
Exchange interaction effects were discovered independently by physicists Werner Heisenberg and Paul Dirac in 1926.
Exchange symmetry Quantum particles are fundamentally indistinguishable. Wolfgang Pauli demonstrated that this is a type of symmetry: states of two particles must be either symmetric or antisymmetric when coordinate labels are exchanged. In a simple one-dimensional system with two identical particles in two states and the system wavefunction can therefore be written two ways:
Exchanging and gives either a symmetric combination of the states ("plus") or an antisymmetric combination ("minus"). Particles that give symmetric combinations are called bosons; those with antisymmetric combinations are called fermions.
The two possible combinations imply different physics. For example, the expectation value of the square of the distance between the two particles is
The last term reduces the expected value for bosons and increases the value for fermions but only when the states and physically overlap
The physical effect of the exchange symmetry requirement is not a force. Rather it is a significant geometrical constraint, increasing the curvature of wavefunctions to prevent the overlap of the states occupied by indistinguishable fermions. The terms "exchange force" and "Pauli repulsion" for fermions are sometimes used as an intuitive description of the effect but this intuition can give incorrect physical results.
Exchange interactions between localized electron magnetic moments Quantum mechanical particles are classified as bosons or fermions. The spin–statistics theorem of quantum field theory demands that all particles with half-integer spin behave as fermions and all particles with integer spin behave as bosons. Multiple bosons may occupy the same quantum state; however, by the Pauli exclusion principle, no two fermions can occupy the same state. Since electrons have spin 1/2, they are fermions. This means that the overall wave function of a system must be antisymmetric when two electrons are exchanged, i.e. interchanged with respect to both spatial and spin coordinates. First, however, exchange will be explained with the neglect of spin.
Exchange of spatial coordinates Taking a hydrogen molecule-like system (i.e. one with two electrons), one may attempt to model the state of each electron by first assuming the electrons behave independently (that is, as if the Pauli exclusion principle did not apply), and taking wave functions in position space of for the first electron and for the second electron. The functions and are orthogonal, and each corresponds to an energy eigenstate. To enforce the indistinguishability of the two electrons, two wave functions for the overall system in position space can be constructed. One uses an antisymmetric combination of the product wave functions in position space:
The other uses a symmetric combination of the product wave functions in position space:
To treat the problem of the hydrogen molecule perturbatively, the overall Hamiltonian is decomposed into an unperturbed Hamiltonian of the non-interacting hydrogen atoms and a perturbing Hamiltonian, which accounts for interactions between the two atoms . The full Hamiltonian is then:
where and
The first two terms of denote the kinetic energy of the electrons. The remaining terms account for attraction between the electrons and their host protons (). The terms in account for the potential energy corresponding to: proton–proton repulsion (), electron–electron repulsion (), and electron–proton attraction between the electron of one host atom and the proton of the other (). All quantities are assumed to be real.
Two eigenvalues for the system energy are found:
where the is the spatially symmetric solution and is the spatially antisymmetric solution, corresponding to and respectively. A variational calculation yields similar results. can be diagonalized by using the position–space functions given by Eqs. (1) and (2). In Eq. (3), is the two-site two-electron Coulomb integral (It may be interpreted as the repulsive potential for electron-one at a particular point in an electric field created by electron-two distributed over the space with the probability density , is the overlap integral, and is the exchange integral, which is similar to the two-site Coulomb integral but includes exchange of the two electrons. It has no simple physical interpretation, but it can be shown to arise entirely due to the anti-symmetry requirement. These integrals are given by:
Although in the hydrogen molecule the exchange integral, Eq. (6), is negative, Heisenberg first suggested that it changes sign at some critical ratio of internuclear distance to mean radial extension of the atomic orbital.Derivation of the Heisenberg Hamiltonian , Rebecca Hihinashvili, accessed on line October 2, 2007.Quantum Theory of Magnetism: Magnetic Properties of Materials, Robert M. White, 3rd rev. ed., Berlin: Springer, 2007, section 2.2.7. .The Theory of Electric and Magnetic Susceptibilities, J. H. van Vleck, London: Oxford University Press, 1932, chapter XII, section 76. The detailed calculation, including evaluation of the above integrals with ground state hydrogen atom wave functions, and application of the variational principle to obtain the minimum energy in both cases of the hydrogen molecule for atomic orbitals and the hydrogen radical (i.e. with only one electron) for molecular orbitals can be found in the book of Müller-Kirsten pp. 272-292 (only in the second edition). Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, Harald J.W. Müller-Kirsten, 2nd ed. (2012), World Scientific, ISBN 978-981-4397-73-5.
Inclusion of spin The symmetric and antisymmetric combinations in Equations (1) and (2) did not include the spin variables (α = spin-up; β = spin-down); there are also antisymmetric and symmetric combinations of the spin variables:
To obtain the overall wave function, these spin combinations have to be coupled with Eqs. (1) and (2). The resulting overall wave functions, called spin-orbitals, are written as Slater determinants. When the orbital wave function is symmetrical, the spin wave function must be anti-symmetrical and vice versa. Accordingly, above corresponds to the spatially symmetric/spin-singlet solution and to the spatially antisymmetric/spin-triplet solution.
J. H. Van Vleck presented the following analysis:Van Vleck, J. H.: Electric and Magnetic Susceptibilities, Oxford, Clarendon Press, p. 318 (1932).
Dirac pointed out that the critical features of the exchange interaction could be obtained in an elementary way by neglecting the first two terms on the right-hand side of Eq. (9), thereby considering the two electrons as simply having their spins coupled by a potential of the form:
It follows that the exchange interaction Hamiltonian between two electrons in orbitals and can be written in terms of their spin momenta and . This interaction is named the Heisenberg exchange Hamiltonian or the Heisenberg–Dirac Hamiltonian in the older literature:
is not the same as the quantity labeled in Eq. (6). Rather, , which is termed the exchange constant, is a function of Eqs. (4), (5), and (6), namely,
However, with orthogonal orbitals (in which = 0), for example with different orbitals in the same atom, .
Effects of exchange If is positive the exchange energy favors electrons with parallel spins; this is a primary cause of ferromagnetism in materials in which the electrons are considered localized in the Heitler–London model of chemical bonding, but this model of ferromagnetism has severe limitations in solids (see below). If is negative, the interaction favors electrons with antiparallel spins, potentially causing antiferromagnetism. The sign of is essentially determined by the relative sizes of and the product of . This sign can be deduced from the expression for the difference between the energies of the triplet and singlet states, :
Although these consequences of the exchange interaction are magnetic in nature, the cause is not; it is due primarily to electric repulsion and the Pauli exclusion principle. In general, the direct magnetic interaction between a pair of electrons (due to their electron magnetic moments) is negligibly small compared to this electric interaction.
Exchange energy splittings are very elusive to calculate for molecular systems at large internuclear distances. However, analytical formulae have been worked out for the hydrogen molecular ion (see references herein).
Normally, exchange interactions are very short-ranged, confined to electrons in orbitals on the same atom (intra-atomic exchange) or nearest neighbor atoms (direct exchange) but longer-ranged interactions can occur via intermediary atoms and this is termed superexchange.
Direct exchange interactions in solids In a crystal, generalization of the Heisenberg Hamiltonian in which the sum is taken over the exchange Hamiltonians for all the pairs of atoms of the many-electron system gives:.
The 1/2 factor is introduced because the interaction between the same two atoms is counted twice in performing the sums. Note that the in Eq.(14) is the exchange constant above not the exchange integral . The exchange integral is related to yet another quantity, called the exchange stiffness constant () which serves as a characteristic of a ferromagnetic material. The relationship is dependent on the crystal structure. For a simple cubic lattice with lattice parameter ,
For a body-centered cubic lattice,
and for a face-centered cubic lattice,
The form of Eq. (14) corresponds identically to the Ising model of ferromagnetism except that in the Ising model, the dot product of the two spin angular momenta is replaced by the scalar product . The Ising model was invented by Wilhelm Lenz in 1920 and solved for the one-dimensional case by his doctoral student Ernst Ising in 1925. The energy of the Ising model is defined to be:
Limitations of the Heisenberg Hamiltonian and the localized electron model in solids